On the homotopy type of certain cobordism categories of surfaces

2010/08/30 by Raptis, George
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1008.5089

Abstract

Let A_g,d be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either S¹ × I with one incoming and one outgoing boundary component or the surface Σ_g,d of genus g and d boundary components that are all incoming. In this paper, we study the homotopy type of the classifying space of the cobordism categories A_g,d and the associated (ordinary) cobordism categories of their connected components \mathbbA_d. A_0,2 is the cobordism category of complex annuli that was considered by Costello and \mathbbA₂ is homotopy equivalent with the positive boundary 1-dimensional embedded cobordism category of Galatius-Madsen-Tillmann-Weiss. We identify their homotopy type with the infinite loop spaces associated with certain Thom spectra.

Related

• The homotopy type of the Poincaré cobordism category for surfaces
• Cobordism categories of manifolds with corners
• Cobordism categories of manifolds with Baas-Sullivan singularities, Part 2
• The homotopy type of the topological cobordism category
• Representations of the homotopy surface category of a simply connected space